[Paper Review] On the Dimension of Finite Point Sets II. "Das Budapester Programm"

György Elekes|arXiv (Cornell University)|Sep 3, 2011

Computational Geometry and Mesh Generation3 references4 citations

TL;DR

This paper introduces 'Das Budapester Programm,' a framework for studying the maximum number of transformations from a group $ G $ that map a large fraction of a finite, properly $ d $-dimensional point set $ \mathcal{P} \subset \mathbb{R}^d $ to itself. It establishes a linear upper bound—$ O(N) $—on the number of such affine transformations in $ \mathbb{R}^2 $ when $ |\varphi(\mathcal{P}) \cap \mathcal{P}| \geq cN $, under the condition that $ \mathcal{P} $ is proper 2D up to a constant factor, using geometric incidence theory and combinatorial decomposition techniques.

ABSTRACT

We are going to use the nickname "Das Budapester Programm" for a large class of finite combinatorial problems to be posed below. Following Klein's original idea, they concern various dimensional Euclidean spaces and several groups of transformations.

Motivation & Objective

• To investigate the maximum number of group transformations (e.g., affine, isometric) that map a large fraction of a finite point set in $ \mathbb{R}^d $ to itself.
• To extend Klein’s Erlanger Programm to finite combinatorial geometry by studying transformation groups acting on discrete point sets.
• To establish sharp upper bounds on the number of such transformations, particularly for the affine group in $ \mathbb{R}^2 $, under proper dimensionality conditions.
• To resolve the 'meta-problem' of whether such bounds scale as $ O(N^\alpha / k^{\alpha-1}) $ for $ \alpha > 1 $, with a focus on linear bounds when $ k = cN $.

Proposed method

• Define a point set as 'proper $ d $-dimensional up to constant $ C $' if it can be partitioned into singletons using $ O(\sqrt[d]{N}) $ hyperplanes.
• Represent each affine transformation $ \varphi \in \text{Aff}(\mathbb{R}^2) $ as a plane $ S_\varphi \subset \mathbb{R}^4 $, so that $ |\varphi(\mathcal{P}) \cap \mathcal{P}| \geq cN $ iff $ |S_\varphi \cap (\mathcal{P}_1 \times \mathcal{P}_2)| \geq cN $, i.e., the plane is $ cN $-rich.
• Use a triple system $ \Delta $ of point-plane incidences to apply combinatorial lemmata: Lemma 5.10 to refine the set of rich planes and their incidences.
• Apply Lemma 5.7 to find a linear number of 'small' triangles (in terms of distance in an arrangement) within each rich plane, ensuring a lower bound on the number of such triangles per plane.
• Bound the total number of such triangles from above using the local cell structure of the arrangement, leveraging the proper dimensionality condition to control neighborhood sizes.
• Derive a contradiction unless the number of rich planes is $ O(N) $, proving the key linear bound via double counting and parameter control.

Experimental results

Research questions

• RQ1 Is the number of affine transformations $ \varphi \in \text{Aff}(\mathbb{R}^2) $ satisfying $ |\varphi(\mathcal{P}) \cap \mathcal{P}| \geq c|\mathcal{P}| $ bounded by $ O(|\mathcal{P}|) $ for any proper 2D point set $ \mathcal{P} $?
• RQ2 Can the meta-problem—bounding $ f_G^d(N,k) = O(N^\alpha / k^{\alpha-1}) $ for $ \alpha > 1 $—be resolved with linear bounds when $ k = cN $?
• RQ3 Does the existence of a 'rich plane' in $ \mathbb{R}^4 $, corresponding to a transformation preserving a large subset of $ \mathcal{P} $, imply structural constraints on the point set via geometric incidence?

Key findings

• For any finite, proper 2D point set $ \mathcal{P} \subset \mathbb{R}^2 $ up to a constant factor $ C $, the number of affine transformations $ \varphi $ such that $ |\varphi(\mathcal{P}) \cap \mathcal{P}| \geq c|\mathcal{P}| $ is at most $ C_1 \cdot |\mathcal{P}| $, where $ C_1 $ depends only on $ C $ and $ c $.
• The bound is linear in $ N = |\mathcal{P}| $, matching the one-dimensional case and confirming the meta-problem’s conjectured scaling for $ \alpha = 1 $, though the general $ \alpha > 1 $ case remains open.
• The proof relies on a geometric incidence argument: rich planes in $ \mathbb{R}^4 $ correspond to transformations preserving many points, and their number is bounded via triangle counting in combinatorial arrangements.
• The method uses a two-step refinement: first, a subset of $ c^*N $-rich planes is selected via Lemma 5.10, then each such plane is shown to contain $ \Omega(N) $ small triangles via Lemma 5.7.
• The total number of such triangles is bounded above by $ (C^*)^2 \varrho^4 N^2 $, where $ \varrho $ depends on the dimensionality constant $ C $, leading to a contradiction unless the number of rich planes is $ O(N) $.
• The result extends in principle to $ \mathbb{R}^3 $ and higher dimensions, provided a technical problem (Problem 5.2) on neighborhood control is resolved.